In a series connection the relationship between the voltage of the capacitor and resistor can be described as follows.
\[U_{cap} + U_R = 0\]Using Ohm’s Law \(R = \frac{U}{I}\) we can substitute for \(U_R\).
\[U_{cap} + IR = 0\] \[U_{cap} = -IR\]We can substitute the electric current with its definition \(I = \frac{dQ}{dt}\).
\[U_{cap} = -\frac{dQ}{dt} * R\]The voltage of the capacitor has the following relationship \(U = \frac{Q}{C}\).
\[\frac{Q}{C} = -\frac{dQ}{dt} * R\]Now we can define a integral with \(Q_0, Q_1\) as the charge range and \(t\) as the discharge duration.
\[\frac{dQ}{Q} = -\frac{dt}{CR}\] \[\int^{Q_1}_{Q_0}\frac{1}{Q} \, dQ = -\int^{t}_{0s}\frac{1}{CR} \, dt\] \[\left[\ln Q + k\right]^{Q_1}_{Q_0} = -\frac{1}{CR}\left[t + k\right]^{t}_{0s}\] \[\ln Q_1 - \ln Q_0 = -\frac{1}{CR} * t\] \[\ln\frac{Q_1}{Q_0} = -\frac{t}{CR}\] \[\frac{Q_1}{Q_0} = e^{-\frac{t}{CR}}\] \[Q_1 = Q_0e^{-\frac{t}{CR}}\]For the charge we can substitute it with the formula for capacitance \(Q = UC\). Remember \(Q_0\) is the charge at the beginning of the discharge, so we can substitute it with the capacitor voltage of the beginning or voltage at max charge.